Weierstrass preparation theorem

The following theorem is known as the Weierstrass preparation theorem, though sometimes that name is reserved for the corollary and this theorem is then known as the Weierstrass division theorem.

In the following we use the standard notation for coordinates in ${ℂ}^n$ that $z=(z_1,\mathrm{\dots },z_n)=(z^{\prime },z_n)$. That is $z^{\prime }$ is the first $n-1$ coordinates.

Theorem.

Let $f\mathrm{:}{\mathrm{C}}^n\mathrm{\to }\mathrm{C}$ be a function analytic in a neighbourhood $U$ of the origin such that $\frac{f\mathit{}\mathrm{(}\mathrm{0}\mathrm{,}{z}_n\mathrm{)}}{z_n^m}$ extends to be analytic at the origin and is not zero at the origin for some positive integer $m$ (in other words, as a function of $z_n$, the function has a zero of order $m$ at the origin). Then there exists a polydisc $D\mathrm{\subset }U$ such that every function $g$ holomorphic and bounded in $D$ can be written as

$$g=qf+r,$$

where $q$ is an analytic function and $r$ is a polynomial in the $z_n$ variable of degree less then $m$ with the coefficients being holomorphic functions in $z^{\mathrm{\prime }}$. Further there exists a constant $C$ independent of $g$ such that

$$\underset{z\in D}{sup}|q(z)|\le C\underset{z\in D}{sup}|g(z)|.$$

The representation $g\mathrm{=}q\mathit{}f\mathrm{+}r$ is unique. Finally the coefficients of the power series expansions of $q$ and $r$ are finite linear combinations of the coefficients of the power series of $g$.

Note that $r$ is not necessarily a Weierstrass polynomial.

Corollary.

Let $f$ be as above, then there is a unique representation of $f$ as $f\mathrm{=}h\mathit{}W$, where $h$ is analytic in a neighbourhood of the origin and $h\mathit{}\mathrm{(}\mathrm{0}\mathrm{)}\mathrm{\ne }\mathrm{0}$ and $W$ being a Weierstrass polynomial.

It should be noted that the condition that $\frac{f(0,z_n)}{z_n^m}$ extends to be analytic, which is equivalent to saying that $f(0,z_n)\not\equiv 0$, is not an essential restriction. In fact $f(0,z_n)\equiv 0$, then there exists a linear change of coordinates, arbitrarily close to the identity, such that the condition of the theorem is satisfied in the new set of coordinates.

Title	Weierstrass preparation theorem
Canonical name	WeierstrassPreparationTheorem
Date of creation	2013-03-22 15:04:28
Last modified on	2013-03-22 15:04:28
Owner	jirka (4157)
Last modified by	jirka (4157)
Numerical id	6
Author	jirka (4157)
Entry type	Theorem
Classification	msc 32B05
Synonym	Weierstrass division theorem
Related topic	WeierstrassPolynomial